Summary: Edge-colorings avoiding rainbow and monochromatic subgraphs
Maria Axenovich and Perry Iverson
Department of Mathematics
Iowa State University
USA
axenovic@math.iastate.edu, piverson@iastate.edu
June 1, 2006
Abstract
For two graphs G and H, let the mixed anti-Ramsey numbers, maxR(n; G, H), (minR(n; G, H))
be the maximum (minimum) number of colors used in an edge-coloring of a complete graph with n
vertices having no monochromatic subgraph isomorphic to G and no totally multicolored (rainbow) sub-
graph isomorphic to H. These two numbers generalize the classical anti-Ramsey and Ramsey numbers,
respectively.
We show that maxR(n; G, H), in most cases, can be expressed in terms of vertex arboricity of H
and it does not depend on the graph G. In particular, we determine maxR(n; G, H) asymptotically for
all graphs G and H, where G is not a star and H has vertex arboricity at least 3.
In studying minR(n; G, H) we primarily concentrate on the case when G = H = K3. We
find minR(n; K3, K3) exactly, as well as all extremal colorings. Among others, by investigating
minR(n; Kt, K3), we show that if an edge-coloring of Kn in k colors has no monochromatic Kt and
no rainbow triangle, then n 2kt2